Question

If in a hyperbola the eccentricity is $\sqrt{3}$, and the distance between the foci is $9$ then the equation of the hyperbola in the standard form is

• A. $\frac{x^2}{{\left(\frac{\sqrt{3}}{2}\right)}^2}-\frac{y^2}{{\left(\sqrt{\frac{3}{2}}\right)}^2}=1$
• B. $\frac{x^2}{{\left(\frac{3\sqrt{3}}{2}\right)}^2}-\frac{y^2}{{\left(\frac{3\sqrt{3}}{\sqrt{2}}\right)}^2}=1$
• C. $\frac{x^2}{{\left(\frac{3\sqrt{3}}{\sqrt{2}}\right)}^2}-\frac{y^2}{{\left(\frac{3\sqrt{2}}{2}\right)}^2}=1$
• D. none of these

Answer 1

Answer: (B)

$\frac{x^2}{{\left(\frac{3\sqrt{3}}{2}\right)}^2}-\frac{y^2}{{\left(\frac{3\sqrt{3}}{\sqrt{2}}\right)}^2}=1$

Given eccentricity of the hyperbola $e=\sqrt{3}$
and distance between focii is 9. $\Rightarrow 2ae=9\Rightarrow a=\frac{3\sqrt{3}}{2}$
also $b^2=a^2(e^2-1)=\frac{27}{4}(3-1)=\frac{27}{2}\Rightarrow b=\frac{3\sqrt{3}}{\sqrt{2}}$
Hence equation of required hyperbola is,
$\frac{x^2}{{\left(\frac{3\sqrt{3}}{2}\right)}^2}-\frac{y^2}{{\left(\frac{3\sqrt{3}}{\sqrt{2}}\right)}^2}=1$
Hence, option 'B' is correct.